Robert F. Nau

Research Summary

My research concerns the mathematical foundations of decision
theory, especially the modeling of rational choice under uncertainty.
The standard paradigm of rational choice is the theory of subjective
expected utility, which makes assumptions about the consistency
and determinacy of behavior by an ideal economic agent. These
assumptions imply that the agent's beliefs are represented by
numerical probabilities, her preferences are represented by numerical
utilities, and her decision-making objective is to maximize the
expected value of utility. This model of rationality underlies
Bayesian methods of inference and decision analysis and much of
mathematical economics, especially microeconomic models of games
and markets.

After decades of extensions, refinements, applications, and empirical
tests of this paradigm, many interesting and important issues
still remain unresolved or controversial. For example, normative
and behavioral decision theorists continue to debate the scope
and limits of rational choice theory; decision analysts and game
theorists continue to dispute the distinction between uncertainty
about states of nature and uncertainty about actions of a rational
opponent; and a variety of disparate and sometimes contradictory
equilibrium models are used to describe collective behavior in
games and markets. My work focuses on some of these unresolved
issues and seeks to develop a simpler, more cohesive, and more
realistic theory of rational choice.

Several main themes run through my research. One is a search for
unity among models of rational choice in different domains
of application. This has led me to focus on no-arbitrage
(avoidance of sure loss) as the fundamental principle of economic
rationality. No-arbitrage is a "primal" definition of
rationality, formulated in terms of extensive variables such as
quantities of money and commodities. By comparison, expected-utility
maximization and various notions of equilibrium are all "dual"
concepts, formulated in terms of intensive variables such as probabilities,
utilities, and prices. One thrust of my research has been to show
that the primal, no-arbitrage characterization of rationality
provides a more seamless transition from the modeling of single-agent
decisions to the modeling of n-agent games and many-agent markets.
The common thread is that agents, whether acting alone or in large
or small groups, should not create arbitrage opportunities for
an outside observer. In all of these settings, the corresponding
dual requirement is that agents should act "as if" maximizing
an appropriate utility function and also "as if" they
had implemented an appropriate equilibrium. The primal approach
has the advantages that it refers to first-order observable quantities
and hence is more robust against problems of measurement (see
the second theme below) and it is inherently a system-level property
and so is more robust against the realities of boundedly rational
behavior at the level of the individual agent (see the third theme
below).

A second main theme in my research is a concern with questions
of measurement and communication. How can an agent articulate
her beliefs and preferences in numerical terms which have a direct
material significance? How can the agent credibly reveal those
beliefs and preferences to a disinterested observer or (more importantly)
to another agent with whom she may be in competition? How do agents
arrive at a state of "common knowledge" of each others'
beliefs and preferences, and how precise can we expect this knowledge
to be? To what extent can beliefs be separated from values--and
does it matter? What are the practical implications of "reciprocal
expectations" of rationality in games and markets? If agents
were behaving irrationally, how would we know? What dynamic or
institutional forces drive agents to "equilibrium" positions?

My approach to these questions is to assume that communication
among agents, or between agents and an observer, is mediated by
material transactions, usually involving money. In this respect
I follow in the tradition of de Finetti (1937, 1974), who defined
subjective probability in terms of the odds at which an agent
would bet money on or against the occurrence of an event at the
discretion of an opponent. Such an approach is advantageous for
several reasons. First, the fundamental measurements explicitly
involve two agents: one who offers to bet or trade and another
who may act upon the offer. Thus, transactions among agents enter
the theory at the most primitive level, setting the stage for
models of inter-agent behavior. Second, money plays a distinguished
role as a yardstick for measuring beliefs and values, which parallels
the role it plays in real economic systems. Third, and most importantly,
the core principle of rationality in this setting is simply the
avoidance of sure loss. De Finetti referred to this as coherence,
but it is known elsewhere as the Dutch book argument or
the no-arbitrage principle. De Finetti's approach traditionally
has been used to model beliefs (i.e., probabilities) alone, under
an assumption of constant marginal utility for money. My work
shows that money-based measurements can be applied to preferences
as well as beliefs, that this measurement process can be carried
out in the presence of nonconstant marginal utility for money,
and that it leads to a new synthesis of decision analysis, game
theory, and market theory.

A third theme in my research is an effort to reconcile normative
and behavioral views of rationality. Normative theory models
the decision processes of idealized economic agents, whereas behavioral
theory describes and interprets decision making as it actually
happens in the laboratory and the field. There has long been a
behavioral countermovement in rational choice research, dating
back to work of Herbert Simon and others in the 1950's, but in
the last 15 years or so there has been an explosive growth of
work in this area. Systematic violations of the expected-utility
hypothesis have been demonstrated in a variety of laboratory situations
and corroborated by studies of how decision making typically occurs
in organizations and markets. Real economic agents often appear
to be rule-followers, role-players, imitators, and heuristic problem-solvers
rather than paragons of consistency and numerical optimization.
These results suggest that normative theories should be formulated
so as to allow for the bounded rationality of real economic agents
and the role of institutions in shaping decision-making behavior.
A number of generalized normative theories of "non-expected
utility" have been proposed in recent years, but none has
succeeded particularly well in giving a parsimonious fit to a
wide range of experimental data and at the same time providing
a tractable basis for economic modeling. My own approach has been
to focus on relaxing those assumptions of the standard theory
which are most implausible from a behavioral viewpoint (especially
the completeness, or perfect precision, of beliefs and
preferences) and to focus on methods of measurement and standards
of rationality which are closely related to those found in real
institutions (e.g., the use of money as a medium of communication
and exchange, and the avoidance of arbitrage).

The following sections discuss the contributions of specific papers
in various topic areas.

2. RESEARCH ON GAMES

Game theory generally starts from the assumption that the "rules
of the game" (i.e., the players' utilities for outcomes and
probabilities for states of nature) are common knowledge, and
it then grapples with the problem of simultaneous expected-utility
maximization.

"Thus each participant attempts to maximize a function...
of which he does not control all variables. This is certainly
no maximum problem but a disconcerting mixture of several conflicting
maximum problems. Every participant is guided by another principle
and neither determines all variables which affect his interest.
This kind of problem is nowhere dealt with in classical mathematics.''
(von Neumann and Morgenstern, Theory of Games and Economic
Behavior)

Numerous solution concepts have been proposed for this problem,
of which the most widely used is still Nash's (1951) equilibrium
concept: the players should select pure or independently randomized
strategies which are best replies to each other. Harsanyi (1967)
generalized Nash's concept to the case of incomplete-information
games (where there is uncertainty about states of nature) by assuming
that the players hold a common prior distribution over states,
and his "common prior assumption" has been widely used
in other economic models. Over the last 15 years there has been
considerable interest in refinements of Nash equilibrium (e.g.,
"perfect"' or "sequential"' equilibrium) and
also in coarsenings (e.g., "rationalizability," "correlated
equilibrium," "communication equilibrium"). Aumann
(1987) has argued that correlated equilibrium (his coarsening
of the Nash concept which allows correlated strategies) is "the"
expression of Bayesian rationality in games, but this claim---which
involves an appeal to the common prior assumption---has been controversial.
Meanwhile, decision analysts generally have been skeptical about
any game-theoretic solution concept which circumscribes
the beliefs that one agent is permitted to hold about the actions
of another (Kadane and Larkey 1982).

The paper "Coherent Behavior in Noncooperative Games"
(with Kevin McCardle) shows that, if the rules of the game are
revealed through material (i.e. money-based) measurements in the
spirit of de Finetti, then common knowledge of rationality takes
on the following simple and precise definition: the players should
not expose themselves collectively to a sure loss---i.e.,
they should be jointly coherent. This simple requirement
captures the intuitive idea of an infinite regress of reciprocal
expectations of rationality, namely that the players should not
behave irrationally as individuals, nor bet on each other
to behave irrationally, nor bet on each other to bet on each other
to behave irrationally, and so on. (Significantly, the same requirement---i.e.,
collective avoidance of sure loss, or arbitrage---also characterizes
competitive equilibria in markets. More about this below.) It
is then proved that this requirement is satisfied if and only
if the outcome of the game is one that occurs with positive probability
in a correlated equilibrium of the game. In other words, rationality
requires the players to behave as if they had implemented
Aumann's concept, of which Nash's concept is a special case. This
paper also gives an elementary proof of the existence of correlated
equilibria, which had been a significant unsolved problem in game
theory since Aumann's original (1974) definition of the concept.
(Another elementary existence proof was developed independently
and roughly contemporaneously by Hart and Schmeidler (1989), although
our proof is mathematically simpler and somewhat more intuitive.
It uses a Markov chain argument that has subsequently been adapted
by Myerson (1995) to define the solution concept of "dual
reduction.")

The preceding results are generalized to incomplete-information
games in "Joint Coherence in Games of Incomplete Information",
where they are shown to lead to a correlated generalization of
Harsanyi's Bayesian equilibrium concept (in games without mechanical
communication devices) or to the communication equilibrium concept
(in games with such devices). Harsanyi's assumption of the common
prior is seemingly vindicated--but these results are obtained
under a simplifying assumption of constant utility for money.
If nonconstant utility for money is (realistically) allowed, a
very different picture emerges, as described in the more recent
paper "Arbitrage-Free Correlated Equilibria."
Because of portfolio effects, we may expect agents' utilities
for money to depend on the outcomes of events. The odds at which
they will bet on such events---i.e., their "revealed"'
probabilities under de Finetti's elicitation method---will then
depend on their utilities for money as well as their probabilities.
More precisely, an agent's revealed probability distribution will
be proportional to the product of her true probabilities and her
state-dependent marginal utilities for money. This amalgam of
probabilities and utilities is known as a risk neutral probability
distribution in the literature of asset pricing. Similar distortions
will affect the agents' revealed utilities for outcomes. Therefore,
the apparent common prior, and the apparent equilibrium distribution
supporting a jointly coherent outcome of the game, must be interpreted
as risk neutral distributions, not the true distributions of any
of the players: their true probability distributions will generally
be heterogeneous. When the revelation of the rules of the game
is thus endogenized through side-gambles among risk averse
players, the possibility also arises that they may rewrite the
rules in the process. Through an accumulation of gambles with
each other or an observer, the agents may even strategically decouple
their actions. For example, if risk-averse players are placed
in a game which is zero-sum in terms of monetary payoffs, the
elicitation process may lead to a Pareto superior allocation in
which both players' payoffs are constant. This stylized fact illustrates
that the natural decoupling effects of monetary transactions in
a public market often reduce the need for strategic behavior between
agents.

The preceding results constitute a "dual" theory of
rational strategic choice: rather than viewing rational agents
first-and-foremost as optimizers, they are viewed as avoiders
of sure loss. The two formulations of the agent's problem---optimization
versus avoiding sure loss---are dual to each other in the sense
of the duality theory of linear programming, but the latter formulation
extends much more readily from the single-agent to the multi-agent
case. The very existence of a dual formulation does not become
apparent until the question is asked: how do the rules of the
game become known? How might the players credibly measure each
other's probabilities and utilities? What is the "acid test"
for a mutually rational outcome? The dual formulation of rationality
in the multi-agent case is simply a collective, or market-level,
no-arbitrage condition---essentially the same condition that underlies
models of rational asset pricing in finance and (as will be seen)
competitive allocations in welfare economics. (Actually, the no-arbitrage
version ought to be viewed as "primal" and the equilibrium
concept as "dual," for reasons noted earlier, but in
any case each is the dual of the other.)

And, as in asset pricing models, risk neutral probabilities emerge
as important variables: the "Harsanyi doctrine" of the
common prior is reinterpreted to apply to risk neutral probabilities,
not true probabilities. In this way the objectionable assumption
of homogeneous true beliefs is relaxed, while the main lines of
Harsanyi's and Aumann's equilibrium concepts are retained. These
results also show that there is no essential difference between
uncertainty about actions of nature and uncertainty about actions
of intelligent opponents: rational agents should avoid "making
book"' against themselves in either case.

3. RESEARCH ON MARKETS

The preceding results suggest that common-knowledge and common-prior
assumptions elsewhere in economics might be subject to similar
reinterpretations, which led me to reexamine Aumann's (1976) seminal
paper on "Agreeing to Disagree." Here, Aumann gives
a formal definition of common knowledge (since used by many others)
and shows that when agents hold common prior beliefs and subsequently
receive heterogeneous information leading them to revise those
beliefs, their posterior probabilities cannot be common knowledge
unless they, too, are identical. In other words, agents cannot
pass from a state of common knowledge of homogeneous beliefs to
common knowledge of heterogeneous beliefs. This result is widely
perceived to imply that the receipt of heterogenous information
cannot provide incentives for trade among perfectly rational agents,
because the disclosure of willingness to trade would render posterior
beliefs common knowledge, at which point they would become identical.
Variations on this no-expected-gain-from-trade result have been
proved by Milgrom and Stokey (1982) and many others, and it is
still viewed as problematic for models of securities markets with
asymmetric information.

In "The Incoherence of Agreeing to Disagree,''
I show that no-expected-gain-from-trade results are illusory.
If Aumann's and Milgrom-Stokey's results are re-cast in terms
of material measurements, then the common prior must be interpreted
as a risk neutral distribution and its construction must be viewed
as the outcome of a process of trade, namely the agents'
measurements of each others' beliefs through betting. Thus, to
assume a common prior is to assume the existence of prior
trade, erasing pre-existing differences in apparent beliefs.

As for posterior trade upon receipt of new information, the no-expected-gain-from
trade results are based on a misapplication of Bayes' theorem
as a model of learning over time, and consequent confusion
between "conditional" and "posterior" probabilities.
Aumann's and Milgrom-Stokey's results refer to the agents' conditional
probabilities held today, given (hypothetically) some information
which is expected to arrive tomorrow. These probabilities must
agree, as a condition of joint coherence. However, the agents'
posterior probabilities held tomorrow upon actual receipt
of that information may still differ: beliefs may drift over time,
like prices in a securities market, and this process of drift
may create incentives for renewed trade. In fact, there even will
be incentives to trade contingent claims in order to quantify
the expected volatility of beliefs. (These
results are discussed in a working paper in progress on "The
Volatility of Beliefs.")

"Arbitrage, Rationality, and Equilibrium"
(with Kevin McCardle) develops the theme, suggested by earlier
work noted above, that coherence or no-arbitrage is the
core principle of rationality which unifies decision analysis,
market theory, and game theory. This is the natural law of the
market---if only because of the vigilance of arbitrageurs---and
its effects trickle down to its constituents, giving them the
appearance of expected-utility maximizers. Thus, rationality in
economic systems can be viewed from the top down, rather than
from the bottom up, and only bounded rationality need be assumed
at the agent level. This paper traces the history of the arbitrage
principle through statistics, economics, and finance, and discusses
the role of risk neutral probabilities and the question of market
completeness. New results are presented concerning the relation
between decision analysis and options pricing methods and concerning
the nature of equilibria in securities markets and exchange economies.

In section 5 of this paper, the role of arbitrage arguments in
decision analysis is discussed. Many important business decisions
(e.g., capital budgeting decisions) take place against the backdrop
of a market where projects can be financed and risks can be hedged
by trading securities. A variety of analytic methods are used
in practice to solve such problems. A popular "naive"
form of decision analysis is the method of discounted cash
flows, in which all cash flows are discounted at a fixed rate
which is chosen to reflect either the firm's or the market's attitude
toward risks of a similar nature. Thus, risk preferences are confounded
with time preferences. A more modern approach, which is claimed
to be superior, is to use option pricing methods (a.k.a.
"contingent claim analysis") in which, for each project
under consideration, a portfolio of securities and options yielding
the same cash flows is constructed (where possible), and the project
is judged worthwhile if its cost is less than the price of the
portfolio. In such a case, the firm could conceivably sell the
portfolio short while undertaking the project, thus reaping an
arbitrage profit. A mathematically equivalent "dual"
method is to compute risk neutral probabilities from market prices
for contingent claims and then discount all cash flows at the
market risk-free interest rate and compute expectations according
to the risk neutral probabilities. The paper shows that standard
methods of decision analysis, in which the firm's own subjective
probabilities and utilities are used to select among projects,
yield precisely the same results as option pricing methods when
the decision problem is properly extended by including the real
possibilities for borrowing at market rates and investing in securities.
In this case, the market acts like a "heat sink," pulling
the risk neutral probabilities of the decision maker in line with
those of the market in any optimal decision. The case of complete
markets is emphasized here; the more general case of incomplete
markets is analyzed in a later paper (with Jim Smith) on "Valuing Risky Projects: Option Pricing Theory and Decision Analysis"
(see below).

In section 6 of "Arbitrage, Rationality, and Equilibrium,"
a conjugate relationship is shown to hold between the multivariate
normal distribution, the exponential utility function, and the
quadratic wealth function with respect to the calculation of risk
neutral probabilities. Thus, if the agent's subjective probability
distribution over security returns is multivariate normal, and
her true utility function is exponential, and her wealth is a
quadratic function of the returns, then her risk neutral distribution
remains in the multivariate normal family. Multivariate normal
distributions and exponential utility functions are widely used
in financial economics. The novelty here lies in the explicit
introduction of a quadratic wealth term representing holdings
of nonlinear contingent claims ("quadratic options"),
whose effect is to shift the covariance matrix of the agent's
risk neutral distribution. Using this conjugate relationship,
the conditions for a common risk neutral distribution (i.e., no
arbitrage) are derived, and it is shown that these conditions
are a CAPM-type relation (i.e., the expected excess return on
a security equals its "beta'' times the expected excess return
on the market) in terms of aggregated subjective beliefs and risk
preferences. The aggregation formulas are generalizations of Lintner's
(1969) heterogeneous-expectations CAPM to the case of complete
markets for contingent claims. The significance of this result
is that it shows the CAPM can be given a purely subjective interpretation,
with no reference either to hypothetical "true" means
and covariances or to historical sample statistics. It also suggests
a role for quadratic options in asset pricing models. (The concept
of a quadratic option has subsequently been discussed by Brennan
(1995).)

Section 7 of the paper re-examines the classical welfare theorem
relating a Pareto optimal wealth allocation to the existence of
a competitive price system. It is shown that if agent preferences
are revealed through willingness to trade commodities (one of
which is money), then common knowledge of Pareto optimality
is simply the no-arbitrage principle under another name, and by
the standard duality argument this requires the existence of prices
and marginal utilities with respect to which the existing
allocation of wealth is competitive. In other words, when utilities
are revealed via material measurements, the elimination of arbitrage
opportunities is necessary and sufficient to drive the economy
to a competitive equilibrium. (As in other settings, the fact
that the measurement process involves incremental transfers of
wealth is significant here: trade typically occurs at non-equilibrium
prices.) The rationale for the existence of a competitive equilibrium
has been debated since the time of Walras, and "tatonnement"'
mechanisms and other hypothetical coordination schemes for reaching
equilibrium have been discussed. This result shows that, under
conditions of common knowledge, a competitive equilibrium is merely
what will remain after all the free lunches have been eaten.

4. RESEARCH ON DECISION ANALYSIS

"Coherent Decision Analysis with Inseparable Probabilities and Utilities"
explores in detail the dual relationship between avoiding sure
loss and maximizing expected utility in the case of a decision-analysis
problem faced by a single agent. (This paper also digs a deeper
foundation under the game-theoretic results discussed earlier.)
It shows that, through the myopic acceptance of small monetary
gambles, the agent can gradually reveal everything about her probabilities
and utilities which is needed to determine her expected-utility-maximizing
decision, and furthermore these gambles will expose her to a sure
loss if she fails to choose that decision. Thus, expected-utility-maximizing
behavior in the "grand world" of her original decision
problem corresponds to coherent behavior in the "small world"
of money-based measurements of her beliefs and preferences.

This approach to decision analysis addresses the problem of separating
probabilities from utilities, which has recently received
attention in papers by Kadane and Winkler (1988), Schervish, Seidenfeld,
and Kadane (1990), and Karni and Schmeidler (1993). If we observe
only an agent's material preferences---i.e., if we cannot credibly
obtain purely "intuitive"' judgments of belief---her
probabilities and utilities will generally be confounded. If we
try to elicit her probabilities via money bets, what we observe
are risk neutral probabilities---products of her true probabilities
and her marginal utilities for money. However, my paper shows
that information about utilities can be elicited in a complementary
way, such that the distorting effect of state-dependent marginal
utility for money cancels out when differences in expected
utility between decisions are calculated. Hence, the inability
to separate probability from utility presents no difficulty, in
principle, for either statistical decision theory or economics.

The role of risk neutral probabilities in decision analysis is
also discussed in section 5 of "Arbitrage, Rationality, and Equilibrium"
(as noted earlier) and in more detail in "Valuing Risky Projects: Options Pricing Theory and Decision Analysis."
The latter paper (with Jim Smith) shows that options pricing methods
and standard methods of decision analysis are fully consistent
(when properly carried out) and can be profitably integrated.
In complete markets, the options pricing framework provides a
convenient separation of the grand decision problem into an "investment"
problem (i.e., whether or not to undertake a project at a specified
cost) and a "financing" problem (how to optimally borrow
to pay for the project and/or hedge its risks by investing in
securities). The investment problem can be solved using only market
data, whereas the financing problem normally requires firm-specific
data. However, in cases where markets are incomplete, option pricing
methods do not yield a precise estimate of the value of a project
nor a complete separation of the grand problem into subproblems.
We show that a partial separation result and a simple procedure
for rolling back the decision tree can be obtained in incomplete
markets under restrictions on preferences (essentially time-additive
exponential utility functions).

In "Indeterminate Probabilities on Finite Sets"
and related work, I construct a theory of confidence-weighted
subjective probabilities. This is a model of "second-order
uncertainty" which addresses some questions about the determinacy
of beliefs which have perenially arisen in Bayesian inference
and decision theory. Most persons instinctively feel that some
of their beliefs can be quantified more precisely than others
and that such differences in precision should somehow be taken
into account in any decision analysis or inference based on those
beliefs. Skepticism about the universal precision of subjective
probabilities is widely felt to have hindered the acceptance of
Bayesian methods, and ad hoc methods of sensitivity analysis
are often applied to prior probabilities in practice.

Many decision theorists have tried to generalize the foundations
of subjective probability theory to embrace the intuitive notion
of precision or confidence associated with probabilities. One
approach is to represent beliefs by intervals of probabilities
rather than point-valued probabilities, and a consistent theory
of interval-valued probabilities can be obtained by merely dropping
the axiom of completeness from the standard theory of de
Finetti or Savage. (The completeness axiom requires that, given
any event and any set of odds, an agent must be willing to bet
either on it or against it---or both.) However, this approach
still leaves some questions unanswered. For example, why should
the endpoints of a probability interval themselves be precisely
determined? How should sensitivity analysis be carried out beyond
these endpoints? How should an inconsistent assessment of interval
probabilities be reconciled? A number of researchers have previously
tried, without great success, to develop a nontrivial and axiomatically
sound model of second-order uncertainty to address these issues.

The theory of confidence-weighted probabilities fulfills this
goal. I show that if the axiom of completeness is dropped and
the axiom of transitivity is also weakened---while still holding
on to coherence---then beliefs are described by lower and upper
probabilities qualified by numerical confidence weights. Thus,
for example, an agent might assert with 100% confidence that the
probability of an event is at least 0.5, and with 50% confidence
that it is at least 0.6. In terms of material measurements, this
means that the stake for which she would bet on it at a rate of
0.6 is only half as large as the stake for which she would bet
on it at a rate of 0.5. (A modified version of de Finetti's elicitation
method is applicable here, in which the betting opponent may take
only convex combinations of offered bets rather than arbitrary
non-negative linear combinations.) The confidence-weighted probabilities
describing an agent's belief in an event are summarized by a concave
function on the unit interval, which can be loosely interpreted
as the indicator function of a "fuzzy" probability interval
or as an "epistemic reliability function" in the terminology
of Gardenfors and Sahlin (1982, 1983). In fact, the laws of confidence-weighted
probabilities are quite similar to the laws of fuzzy probabilities
used by some authors, but the model is not based in fuzzy set
theory. Rather, it provides independent support for the idea that
fuzzy sets might be useful for representing imprecise personal
probabilities and expectations (which are subjectively determined
subsets of the real numbers in the first place), but not necessarily
for representing other forms of cognitive imprecision.

The companion paper on "Decision Analysis with Indeterminate or Incoherent Probabilities"
applies confidence-weighted probabilities to the analysis of a
finite-state decision problem and shows in detail how they can
be used to perform sensitivity analysis and to reconcile incoherence.
This line of work shows that the standard Bayesian model of determinate
probabilities and the more general model of interval-valued probabilities
can both be embedded in a more general framework of confidence-weighted
probabilities which is axiomatically sound, which satisfies the
intuitive desiderata for a theory of second-order uncertainty,
and which justifies a pragmatic approach to sensitivity analysis
and the reconciliation of inconsistency in decision-analysis models.

Another potentially interesting application of confidence-weighted
probabilities is to the problem of combining expert judgments.
A well-known impossibility theorem (Genest and Zidek 1986) states
that there can be no formula for combining ordinary probability
judgments from different individuals that preserves consensus
(where it exists) and simultaneously satisfies Bayesian principles
of updating probabilities upon receipt of new information. However,
confidence-weighted probabilities from different individuals
can be combined by a simple linear pooling formula which does
preserve consensus and respect Bayesian updating. In other words,
the confidence-weighted probability model is "closed"
under the operation of combining judgments. The intuitive reason
for this is that the problem of combining judgments inherently
involves tradeoffs among the possibly-conflicting beliefs of several
individuals who may have differing degrees of reliability or expertise.
Such tradeoffs implicitly require some kind of relative weighting
of judgments both within and between individuals. The confidence-weighted
probability model has such a set of weights built-in, and so a
combination of confidence-weighted probability judgments from
different individuals has the same qualitative properties as a
set of judgments from single individual. (Or to put it another
way, the confidence-weighted probability model represents a somewhat
schizophrenic individual whose judgments may have differing degrees
of confidence and may even be inconsistent. Such an individual
can be compared to a roomful of experts with different opinions.)

A more recent paper on "The Shape of Incomplete Preferences"
presents a joint axiomatization of subjective probability and
utility minus the completeness axiom. This result combines
the features of Smith's (1961) theory of interval-valued probabilities
with those of Aumann's (1962) theory of interval-valued utilities,
and provides a foundation for methods of robust Bayesian statistics.
It has close connections to recent work on partially ordered preferences
by Seidenfeld, Schervish, and Kadane (1995) but emphasizes duality
arguments mores strongly.

The paper "Should Scoring Rules be 'Effective'?"
considers the elicitation of probabilities via scoring rules
(reward or penalty functions) rather than bets, and examines a
number of desiderata which have been proposed for such rules.
Characterizations of important classes of scoring rules are given
for both continuous and discrete probability forecasts, and theorems
are proved concerning the relationships between scoring rules
and metrics for measuring distances between probability
distributions. It is argued that the only essential property of
a scoring rule is that of properness (i.e., rewarding honest
reporting of probabilities) and that the choice of a scoring rule
should be tailored, insofar as it is possible, to the decision
problem for which the probability is relevant.

The note "Blau's Dilemma Revisited"
examines the controversy between Bayesian utility maximization
and chance-constrained programming as paradigms of choice under
uncertainty, pointing out some of the shortcomings of the latter.